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Question:
Suppose the fractal dimension $1\le d_c\le2$ of a planar parametric curve $c(t) := (x(t),y(t))$ is given;
can any nontrivial estimates for the fractal dimensions $d_x$ of $x(t)$ and $d_y$ of $y(t)$ be given?

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No nontrivial estimates, at least if the definitions are right.

Say let's define $$\ell_\alpha c=\sup\,\left\{\,\sum_{t_0<\dots<t_n}|c(t_i)-c(t_{i-1})|^\alpha\,\right\}$$ and $$\dim c=\sup\{\,\alpha\mid \ell_\alpha c<\infty\,\}.$$

Then from Pythagorean theorem we get $$\dim c=\max\{\dim x, \dim y\}.$$

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